Definition. A commutative ring with non-zero unity is a Prüfer ring (cf. Prüfer domain) if every finitely generated regular ideal of is invertible. (It can be proved that if every ideal of generated by two elements is invertible, then all finitely generated ideals are invertible; cf. invertibility of regularly generated ideal.)
Theorem 1 (Pahikkala 1982). Let be a commutative ring with non-zero unity and let be the total ring of fractions of . Then, is a Prüfer ring iff the equation
Theorem 2 (Pahikkala 1982). The commutative ring with non-zero unity is Prüfer ring iff the multiplication rule
The proofs are found in the paper
J. Pahikkala 1982: “Some formulae for multiplying and inverting ideals”. – Annales universitatis turkuensis 183. Turun yliopisto (University of Turku).
An additional characterization of Prüfer ring is found here in the entry “least common multiple (http://planetmath.org/LeastCommonMultiple)”, several other characterizations in  (p. 238–239).
Note. A commutative ring satisfying the equation (1) for all polynomials is called a Gaussian ring. Thus any Prüfer domain (http://planetmath.org/PruferDomain) is always a Gaussian ring, and conversely (http://planetmath.org/Converse), an integral domain, which is a Gaussian ring, is a Prüfer domain. Cf. .
- 1 M. Larsen & P. McCarthy: Multiplicative theory of ideals. Academic Press. New York (1971).
- 2 Sarah Glaz: “The weak dimensions of Gaussian rings”. – Proc. Amer. Math. Soc. (2005).
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